///////////////////////////////////////////////////////////////////////////
//
// Copyright (c) 2010, Industrial Light & Magic, a division of Lucas
// Digital Ltd. LLC
//
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///////////////////////////////////////////////////////////////////////////
#include "ImathMatrixAlgo.h"
#include <iostream>
#include <assert.h>
#include <cmath>
#include <limits>
#include <algorithm>
template <typename T>
void
verifyOrthonormal (const IMATH_INTERNAL_NAMESPACE::Matrix33<T>& A)
{
const T valueEps = T(100) * std::numeric_limits<T>::epsilon();
const IMATH_INTERNAL_NAMESPACE::Matrix33<T> prod = A * A.transposed();
for (int i = 0; i < 3; ++i)
{
for (int j = 0; j < 3; ++j)
{
if (i == j)
assert (std::abs (prod[i][j] - 1) < valueEps);
else
assert (std::abs (prod[i][j]) < valueEps);
}
}
}
template <typename T>
void
verifyOrthonormal (const IMATH_INTERNAL_NAMESPACE::Matrix44<T>& A)
{
const T valueEps = T(100) * std::numeric_limits<T>::epsilon();
const IMATH_INTERNAL_NAMESPACE::Matrix44<T> prod = A * A.transposed();
for (int i = 0; i < 4; ++i)
{
for (int j = 0; j < 4; ++j)
{
if (i == j)
assert (std::abs (prod[i][j] - 1) <= valueEps);
else
assert (std::abs (prod[i][j]) <= valueEps);
}
}
}
template <typename T>
void
verifyTinySVD_3x3 (const IMATH_INTERNAL_NAMESPACE::Matrix33<T>& A)
{
T maxEntry = 0;
for (int i = 0; i < 3; ++i)
for (int j = 0; j < 3; ++j)
maxEntry = std::max (maxEntry, std::abs (A[i][j]));
const T eps = std::numeric_limits<T>::epsilon();
const T valueEps = maxEntry * T(10) * eps;
for (int i = 0; i < 2; ++i)
{
const bool posDet = (i == 0);
IMATH_INTERNAL_NAMESPACE::Matrix33<T> U, V;
IMATH_INTERNAL_NAMESPACE::Vec3<T> S;
IMATH_INTERNAL_NAMESPACE::jacobiSVD (A, U, S, V, eps, posDet);
IMATH_INTERNAL_NAMESPACE::Matrix33<T> S_times_Vt;
for (int i = 0; i < 3; ++i)
for (int j = 0; j < 3; ++j)
S_times_Vt[i][j] = S[j] * V[i][j];
S_times_Vt.transpose();
// Verify that the product of the matrices is A:
const IMATH_INTERNAL_NAMESPACE::Matrix33<T> product = U * S_times_Vt;
for (int i = 0; i < 3; ++i)
for (int j = 0; j < 3; ++j)
assert (std::abs (product[i][j] - A[i][j]) <= valueEps);
// Verify that U and V are orthogonal:
if (posDet)
{
assert (U.determinant() > 0.9);
assert (V.determinant() > 0.9);
}
// Verify that the singular values are sorted:
for (int i = 0; i < 2; ++i)
assert (S[i] >= S[i+1]);
// Verify that all the SVs except maybe the last one are positive:
for (int i = 0; i < 2; ++i)
assert (S[i] >= T(0));
if (!posDet)
assert (S[2] >= T(0));
verifyOrthonormal (U);
verifyOrthonormal (V);
}
}
template <typename T>
void
verifyTinySVD_4x4 (const IMATH_INTERNAL_NAMESPACE::Matrix44<T>& A)
{
T maxEntry = 0;
for (int i = 0; i < 4; ++i)
for (int j = 0; j < 4; ++j)
maxEntry = std::max (maxEntry, std::abs (A[i][j]));
const T eps = std::numeric_limits<T>::epsilon();
const T valueEps = maxEntry * T(100) * eps;
for (int i = 0; i < 2; ++i)
{
const bool posDet = (i == 0);
IMATH_INTERNAL_NAMESPACE::Matrix44<T> U, V;
IMATH_INTERNAL_NAMESPACE::Vec4<T> S;
IMATH_INTERNAL_NAMESPACE::jacobiSVD (A, U, S, V, eps, posDet);
IMATH_INTERNAL_NAMESPACE::Matrix44<T> S_times_Vt;
for (int i = 0; i < 4; ++i)
for (int j = 0; j < 4; ++j)
S_times_Vt[i][j] = S[j] * V[i][j];
S_times_Vt.transpose();
// Verify that the product of the matrices is A:
const IMATH_INTERNAL_NAMESPACE::Matrix44<T> product = U * S_times_Vt;
for (int i = 0; i < 4; ++i)
for (int j = 0; j < 4; ++j)
assert (std::abs (product[i][j] - A[i][j]) <= valueEps);
// Verify that U and V have positive determinant if requested:
if (posDet)
{
assert (U.determinant() > 0.99);
assert (V.determinant() > 0.99);
}
// Verify that the singular values are sorted:
for (int i = 0; i < 3; ++i)
assert (S[i] >= S[i+1]);
// Verify that all the SVs except maybe the last one are positive:
for (int i = 0; i < 3; ++i)
assert (S[i] >= T(0));
if (!posDet)
assert (S[3] >= T(0));
verifyOrthonormal (U);
verifyOrthonormal (V);
}
}
template <typename T>
void
testTinySVD_3x3 (const IMATH_INTERNAL_NAMESPACE::Matrix33<T>& A)
{
std::cout << "Verifying SVD for [[" << A[0][0] << ", " << A[0][1] << ", " << A[0][2] << "], "
<< "[" << A[1][0] << ", " << A[1][1] << ", " << A[1][2] << "], "
<< "[" << A[2][0] << ", " << A[2][1] << ", " << A[2][2] << "]]\n";
verifyTinySVD_3x3 (A);
verifyTinySVD_3x3 (A.transposed());
// Try all different orderings of the columns of A:
int cols[3] = { 0, 1, 2 };
do
{
IMATH_INTERNAL_NAMESPACE::Matrix33<T> B;
for (int i = 0; i < 3; ++i)
for (int j = 0; j < 3; ++j)
B[i][j] = A[i][cols[j]];
verifyTinySVD_3x3 (B);
} while (std::next_permutation (cols, cols + 3));
}
template <typename T>
void
testTinySVD_3x3 (const T a, const T b, const T c,
const T d, const T e, const T f,
const T g, const T h, const T i)
{
const IMATH_INTERNAL_NAMESPACE::Matrix33<T> A (a, b, c, d, e, f, g, h, i);
testTinySVD_3x3 (A);
}
template <typename T>
void
testTinySVD_4x4 (const IMATH_INTERNAL_NAMESPACE::Matrix44<T>& A)
{
std::cout << "Verifying SVD for [[" << A[0][0] << ", " << A[0][1] << ", " << A[0][2] << ", " << A[0][3] << "], "
<< "[" << A[1][0] << ", " << A[1][1] << ", " << A[1][2] << ", " << A[1][3] << "], "
<< "[" << A[2][0] << ", " << A[2][1] << ", " << A[2][2] << ", " << A[2][3] << "], "
<< "[" << A[3][0] << ", " << A[3][1] << ", " << A[3][2] << ", " << A[3][3] << "]]\n";
verifyTinySVD_4x4 (A);
verifyTinySVD_4x4 (A.transposed());
// Try all different orderings of the columns of A:
int cols[4] = { 0, 1, 2, 3 };
do
{
IMATH_INTERNAL_NAMESPACE::Matrix44<T> B;
for (int i = 0; i < 4; ++i)
for (int j = 0; j < 4; ++j)
B[i][j] = A[i][cols[j]];
verifyTinySVD_4x4 (B);
} while (std::next_permutation (cols, cols + 4));
}
template <typename T>
void
testTinySVD_4x4 (const T a, const T b, const T c, const T d,
const T e, const T f, const T g, const T h,
const T i, const T j, const T k, const T l,
const T m, const T n, const T o, const T p)
{
const IMATH_INTERNAL_NAMESPACE::Matrix44<T> A (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p);
testTinySVD_4x4 (A);
}
template <typename T>
void
testTinySVDImp()
{
// Try a bunch of 3x3 matrices:
testTinySVD_3x3<T> (1, 0, 0, 0, 1, 0, 0, 0, 1);
testTinySVD_3x3<T> (1, 0, 0, 0, -1, 0, 0, 0, 1);
testTinySVD_3x3<T> (0, 0, 0, 0, 0, 0, 0, 0, 0);
testTinySVD_3x3<T> (0, 0, 0, 0, 0, 0, 0, 0, 1);
testTinySVD_3x3<T> (1, 0, 0, 0, 1, 0, 0, 0, 0);
testTinySVD_3x3<T> (1, 0, 0, 0, 0, 0, 0, 0, 0);
testTinySVD_3x3<T> (1, 0, 0, 1e-10, 0, 0, 0, 0, 0);
testTinySVD_3x3<T> (1, 0, 0, 1e-10, 0, 0, 0, 0, 100000);
testTinySVD_3x3<T> (1, 2, 3, 4, 5, 6, 7, 8, 9);
testTinySVD_3x3<T> (1, 2, 3, 4, 5, 6, 7, 8, 9);
testTinySVD_3x3<T> (outerProduct (IMATH_INTERNAL_NAMESPACE::Vec3<T> (100, 1e-5, 0), IMATH_INTERNAL_NAMESPACE::Vec3<T> (100, 1e-5, 0)));
testTinySVD_3x3<T> (outerProduct (IMATH_INTERNAL_NAMESPACE::Vec3<T> (245, 20, 1), IMATH_INTERNAL_NAMESPACE::Vec3<T> (256, 300, 20)));
testTinySVD_3x3<T> (outerProduct (IMATH_INTERNAL_NAMESPACE::Vec3<T> (245, 20, 1), IMATH_INTERNAL_NAMESPACE::Vec3<T> (245, 20, 1)) +
outerProduct (IMATH_INTERNAL_NAMESPACE::Vec3<T> (1, 2, 3), IMATH_INTERNAL_NAMESPACE::Vec3<T> (1, 2, 3)));
// Some problematic matrices from SVDTest:
testTinySVD_3x3<T> (
0.0023588321752040036, -0.0096558131480729038, 0.0010959850449366493,
0.0088671829608044754, 0.0016771794267033666, -0.0043081475729438235,
0.003976050440932701, 0.0019880497026345716, 0.0089576046614601966);
testTinySVD_3x3<T> (
2.3588321752040035e-09, -9.6558131480729038e-09, 1.0959850449366498e-09,
8.8671829608044748e-09, 1.6771794267033661e-09, -4.3081475729438225e-09,
3.9760504409327016e-09, 1.9880497026345722e-09, 8.9576046614601957e-09);
testTinySVD_3x3<T> (
-0.46673855799602715, 0.67466260360310948, 0.97646986796448998,
-0.032460753747103721, 0.046584527749418278, 0.067431228641151142,
-0.088885055229687815, 0.1280389179308779, 0.18532617511453064);
testTinySVD_3x3<T> (
1e-8, 0, 0,
0, 1e-8, 0,
0, 0, 1e-8);
testTinySVD_3x3<T> (
1, 0, 0,
0, .00036, 0,
1e-18, 0, .00018);
testTinySVD_3x3<T> (
1.3, 0, 0,
0, .0003, 0,
1e-17, 0, 0);
testTinySVD_3x3<T> (
1, 0, 0,
0, 1e-2, 0,
0, 0, 1e-2);
testTinySVD_3x3<T> (
1,0,0,
0,1,0,
0,0,0);
testTinySVD_3x3<T> (
1, 0, 0,
0, 1e-3, 0,
0, 0, 1e-6);
testTinySVD_3x3<T> (
0.59588638570136332, -0.79761234126107794, -1,
0.39194500425202045, 0.91763115383440363, -0.341818175044664,
-0.45056075218951946, -0.71259057727425101, 0.47125008216720271);
testTinySVD_3x3<T> (
4.38805348e-09, -2.53189691e-09, -4.65678607e-09,
-3.23000099e-10, 1.86370294e-10, 3.42781192e-10,
-4.61572824e-09, 2.6632645e-09, 4.89840346e-09);
// problematic 2x2 one for lapack on suse (see below), padded with 0's
testTinySVD_3x3<T> (
0, -1.00000003e-22, 0,
1.00000001e-07, 0, 0,
0, 0, 0);
// problematic 2x2 one for lapack on suse (see below), padded with 0's and 1
testTinySVD_3x3<T> (
0, -1.00000003e-22, 0,
1.00000001e-07, 0, 0,
0, 0, 1);
// Now, 4x4 matrices:
testTinySVD_4x4<T> (1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1);
testTinySVD_4x4<T> (1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1);
testTinySVD_4x4<T> (1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0);
testTinySVD_4x4<T> (1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0);
testTinySVD_4x4<T> (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0);
testTinySVD_4x4<T> (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0);
testTinySVD_4x4<T> (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16);
testTinySVD_4x4<T> (0, -1.00000003e-22, 0, 0, 00000001e-07, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1);
testTinySVD_4x4<T> (outerProduct (IMATH_INTERNAL_NAMESPACE::Vec4<T> (100, 1e-5, 0, 0), IMATH_INTERNAL_NAMESPACE::Vec4<T> (100, 1e-5, 0, 0)));
testTinySVD_4x4<T> (outerProduct (IMATH_INTERNAL_NAMESPACE::Vec4<T> (245, 20, 1, 0.5), IMATH_INTERNAL_NAMESPACE::Vec4<T> (256, 300, 20, 10)));
testTinySVD_4x4<T> (outerProduct (IMATH_INTERNAL_NAMESPACE::Vec4<T> (245, 20, 1, 0.5), IMATH_INTERNAL_NAMESPACE::Vec4<T> (256, 300, 20, 10)) +
outerProduct (IMATH_INTERNAL_NAMESPACE::Vec4<T> (30, 10, 10, 10), IMATH_INTERNAL_NAMESPACE::Vec4<T> (1, 2, 3, 3)));
}
void
testTinySVD ()
{
std::cout << "Testing TinySVD algorithms in single precision..." << std::endl;
testTinySVDImp<float>();
std::cout << "Testing TinySVD algorithms in double precision..." << std::endl;
testTinySVDImp<double>();
}